Optimality

314 D. Bhattacharya and A. K. Basu
Throughout the paper the following notation is used: φy(µ, σ 2 ) represents the normal
density with mean µ and variance σ 2 ; the symbol ‘=⇒’ denotes convergence
in distribution, and the symbol ‘→’ denotes convergence in Pθ0,n probability.
Now let the sequence of statistical experiments En ={Xn,An, Pθ,n}n≥1 be a
locally asymptotically mixture of normals (LAMN) at θ0∈ Θ. For the definition of
a LAMN experiment the reader is referred to Bhattacharya and Roussas [6]. Then
there exist random variables Zn and Wn (Wn > 0 a.s.) such that
(2.1)
and
(2.2)
Λn(θ0, θn) = log dPθ0+δnh,n
dPθ0,n
− hZn + 1
2 h2 Wn→ 0,
(Zn, Wn)⇒(Z, W) under Pθ0,n,
where Z = W 1/2 G, G and W are independently distributed, W > 0 a.s. and
G∼N(0,1). Moreover, the distribution of W does not depend on the parameter h
(Le Cam and Yang [13]).
The following examples illustrate the different quantities appearing in equations
(2.1) and (2.2) and in the subsequent derivations.
Example 2.1 (An explosive autoregressive process of first order). Let the
random variables Xj, j = 1,2, . . . satisfy a first order autoregressive model defined
by
(2.3) Xj = θXj−1 + ɛj, X0 = 0,|θ| > 1,
where ɛj’s are i.i.d. N(0,1) random variables. We consider the explosive case where
|θ| > 1. For this model we can write
fj(θ) = f(xj|x1, . . . , xj−1;θ) ∝ e
1 2
− 2
(xj−θxj−1)
.
Let θ0 be the true value of θ. It can be shown that for the model described in
(2.3) we can select the sequence of norming constants δn = (θ2 0−1) θn so that (2.1) and
0
(2.2) hold. Clearly δn→ 0 as n→∞. We can also obtain Wn(θ0), Zn(θ0) and their
asymptotic distributions, as n→∞, as follows:
Wn(θ0) = (θ2 0− 1) 2
θ 2n
0
Gn(θ0) = (
n�
j=1
n�
j=1
X 2 1 − 2
j−1) (
X 2 j−1⇒ W as n→∞, where W∼ χ 2 1 and
n�
n�
Xj−1ɛj) = (
j=1
where G∼N(0,1) and ˆ θn is the m.l.e. of θ. Also
Zn(θ0) = W 1
2
θ n 0
j=1
n (θ0)Gn(θ0) = (θ2 n�
0− 1)
(
where W is independent of G. It also holds that
j=1
(Zn(θ0), Wn(θ0))⇒(Z, W).
X 2 j−1) 1
2 ( ˆ θn− θ)⇒G,
Xj−1ɛj)⇒W 1
2 G = Z,
Hence Z|W∼ N(0, W). In general Z is a mixture of normal distributions with W
as the mixing variable.